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Difference between revisions of "2008 AIME II Problems/Problem 7"

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__TOC__
 
== Problem ==
 
== Problem ==
 
Let <math>r</math>, <math>s</math>, and <math>t</math> be the three roots of the equation
 
Let <math>r</math>, <math>s</math>, and <math>t</math> be the three roots of the equation
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Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>.
 
Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>.
  
−
__TOC__
+
==Video Solution by Punxsutawney Phil==
−
== Solution ==
+
https://www.youtube.com/watch?v=6mYZYh9gJBs
−
=== Solution 1 ===
+
 
 +
==Solution 1 ==
 +
By [[Vieta's formulas]], we have <math>r + s + t = 0</math> so <math>t = -r - s.</math> Substituting this into our problem statement, our desired quantity is <cmath>(r + s)^3 - r^3 - s^3 = 3r^2s + 3rs^2 = 3rs(r + s).</cmath> Also by [[Vieta's formulas]] we have <cmath>rst = -rs(r + s) = -\dfrac{2008}{8} = -251</cmath> so negating both sides and multiplying through by 3 gives our answer of <math>\boxed{753}.</math>
 +
 
 +
== Solution 2 ==
 
By [[Vieta's formulas]], we have <math>r+s+t = 0</math>, and so the desired answer is <math>(r+s)^3 + (s+t)^3 + (t+r)^3 = (0-t)^3 + (0-r)^3 + (0-s)^3 = -(r^3 + s^3 + t^3)</math>. Additionally, using the factorization
 
By [[Vieta's formulas]], we have <math>r+s+t = 0</math>, and so the desired answer is <math>(r+s)^3 + (s+t)^3 + (t+r)^3 = (0-t)^3 + (0-r)^3 + (0-s)^3 = -(r^3 + s^3 + t^3)</math>. Additionally, using the factorization
 
<cmath>r^3 + s^3 + t^3 - 3rst = (r+s+t)(r^2 + s^2 + t^2 - rs - st - tr) = 0</cmath>
 
<cmath>r^3 + s^3 + t^3 - 3rst = (r+s+t)(r^2 + s^2 + t^2 - rs - st - tr) = 0</cmath>
 
we have that <math>r^3 + s^3 + t^3 = 3rst</math>. By Vieta's again, <math>rst = \frac{-2008}8 = -251 \Longrightarrow -(r^3 + s^3 + t^3) = -3rst = \boxed{753}.</math>
 
we have that <math>r^3 + s^3 + t^3 = 3rst</math>. By Vieta's again, <math>rst = \frac{-2008}8 = -251 \Longrightarrow -(r^3 + s^3 + t^3) = -3rst = \boxed{753}.</math>
  
−
=== Solution 2 ===
+
== Solution 3 ==
 
Vieta's formulas gives <math>r + s + t = 0</math>. Since <math>r</math> is a root of the polynomial, <math>8r^3 + 1001r + 2008 = 0\Longleftrightarrow - 8r^3 = 1001r + 2008</math>, and the same can be done with <math>s,\ t</math>. Therefore, we have
 
Vieta's formulas gives <math>r + s + t = 0</math>. Since <math>r</math> is a root of the polynomial, <math>8r^3 + 1001r + 2008 = 0\Longleftrightarrow - 8r^3 = 1001r + 2008</math>, and the same can be done with <math>s,\ t</math>. Therefore, we have
 
<cmath>\begin{align*}8\{(r + s)^3 + (s + t)^3 + (t + r)^3\} &= - 8(r^3 + s^3 + t^3)\\
 
<cmath>\begin{align*}8\{(r + s)^3 + (s + t)^3 + (t + r)^3\} &= - 8(r^3 + s^3 + t^3)\\
−
&= 1001(r + s + t) + 2008\cdot 3 = 3\cdot 2008\end{align*}</cmath>yielding the answer <math>753</math>.
+
&= 1001(r + s + t) + 2008\cdot 3 = 3\cdot 2008\end{align*}</cmath>yielding the answer <math>\boxed{753}</math>.
 +
 
 +
Also, Newton's Sums yields an answer through the application.
 +
http://www.artofproblemsolving.com/Wiki/index.php/Newton's_Sums
 +
 
 +
== Solution 4 ==
 +
Expanding, you get:
 +
<cmath>r^3 + 3r^2s + 3s^2r +s^3 +</cmath>
 +
<cmath>s^3 + 3s^2t + 3t^2s +t^3 +</cmath>
 +
<cmath>r^3 + 3r^2t + 3t^2r +t^3</cmath>
 +
<cmath>= 2r^3 + 2s^3 + 2t^3 + 3r^2s + 3s^2r + 3s^2t + 3t^2s + 3r^2t + 3t^2r </cmath>
 +
This looks similar to <math>(r+s+t)^3 = r^3 + s^3 + t^3 + 3r^2s + 3s^2r + 3s^2t + 3t^2s + 3r^2t + 3t^2r + 6rst</math>
 +
Substituting:
 +
<cmath>(r+s+t)^3 - 6rst + r^3+s^3+t^3 = (r + s)^3 + (s + t)^3 + (t + r)^3</cmath>
 +
Since <math>r+s+t = 0</math>,
 +
<cmath>(r+s)^3 + (s+t)^3 + (t+r)^3 = (0-t)^3 + (0-r)^3 + (0-s)^3 = -(r^3 + s^3 + t^3)</cmath>
 +
Substituting, we get <cmath>(r+s+t)^3 - 6rst + r^3+s^3+t^3 =  -(r^3 + s^3 + t^3)</cmath>
 +
or, <cmath>0^3 - 6rst + r^3+s^3+t^3 =  -(r^3 + s^3 + t^3) \implies 2(r^3 + s^3 + t^3) = 6rst</cmath>
 +
We are trying to find <math> -(r^3 + s^3 + t^3)</math>.
 +
Substituting:
 +
<cmath> -(r^3 + s^3 + t^3) = -3srt = \frac{-2008*3}{8} = \boxed{753}.</cmath>
 +
 
 +
== Solution 5 ==
 +
Write <math>(r+s)^3+(s+t)^3+(t+r)^3=-(r^3+s^3+t^3)</math> and let <math>f(x)=8x^3+1001x+2008</math>. Then <cmath>f(r)+f(s)+f(t)=8(r^3+s^3+t^3)+1001(r+s+t)+6024=8(r^3+s^3+t^3)+6024=0.</cmath> Solving for <math>r^3+s^3+t^3</math> and negating the result yields the answer <math>\boxed{753}.</math>
 +
 
 +
== Solution 6 ==
 +
Here by [[Vieta's formulas]]:
 +
<math>r+s+t = 0</math> --(1)
 +
 
 +
<math>rst = \frac{-2008}{8} = -251</math> --(2)
 +
 
 +
By the factorisation formula:
 +
Let <math>a = r+s</math>, <math>b = s+t</math>, <math>c = t+r</math>,
 +
<math>a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = 0</math> (By (1))
 +
 
 +
So <cmath>a^3+b^3+c^3 = 3abc = 3(r+s)(s+t)(t+r) = 3(-t)(-r)(-s) = 3[-(-251)] = \boxed{753}.</cmath>
 +
 
 +
== Solution 7 ==
 +
Let's construct a polynomial with the roots <math>(r+s), (s+t),</math> and <math>(t+r)</math>.
 +
 
 +
sum of the roots:
 +
 
 +
<math>=2(r+s+t)=2\cdot0=0</math>
 +
 
 +
pairwise product of the roots:
 +
 
 +
<math>(r+s)(s+t)+(s+t)(t+r)+(t+r)(r+s)=r^2+s^2+t^2+3(rs+st+tr)</math>
 +
 
 +
<math>=(r+s+t)^2+rs+st+tr=0+\frac{1001}{8}</math>
 +
 
 +
product of the roots:
 +
 
 +
<math>(r+s)(s+t)(t+r)=r^2t+r^2s+s^2r+s^2t+t^2r+t^2s+3rst</math>
 +
 
 +
<math>=(rs+st+tr)(r+s+t)-3rst+2rst=-rst=-\frac{2008}{8}</math>
 +
 
 +
thus, the polynomial we get is
 +
 
 +
<math>x^3+\frac{1001}{8}x+-\frac{2008}{8}=0</math>
 +
 
 +
as <math>(r+s), (s+t),</math> and <math>(t+r)</math> are roots of this polynomial, we know that (using power reduction)
 +
 
 +
<math>(r+s)^3+\frac{1001}{8}(r+s)-\frac{2008}{8}=0</math>
 +
 
 +
<math>(s+t)^3+\frac{1001}{8}(s+t)-\frac{2008}{8}=0</math>
 +
 
 +
<math>(t+r)^3+\frac{1001}{8}(t+r)-\frac{2008}{8}=0</math>
 +
 
 +
adding all of the equations up, we see that
 +
 
 +
<math>(r+s)^3+(s+t)^3+(t+r)^3=3\cdot\frac{2008}{8}-\frac{1001}{8}(2r+2s+2t)=251(3)+0=\boxed{753}</math>
 +
 
 +
==Solution 8==
 +
 
 +
We want to find what is <math>-(r^3+s^3+t^3)</math> which reminds us of Newton sum. So we can see that <math>8S_3+0\cdot S_2+1001\cdot S_1+3\cdot 2008=0</math> Notice that <math>S_1=0</math> so it is just <math>S_3=-\frac{2008\cdot 3}{8}=-753</math>, the desired answer is <math>\boxed{753}</math>
 +
 
 +
~bluesoul
 +
 
 +
==Solution 9==
 +
This solution uses Vietas, as with everyone else's solution.
 +
Expanding the expression we get
 +
 
 +
<cmath>(r+s)^3+(s+t)^3+(t+r)^3 = r^3+3r^2s+3rs^2+\dots +3s^2t+3ts^2+t^3</cmath>
 +
 
 +
Seeing the cubes, we try to find a <math>(r+s+t)^3</math> and upon doing so, we get
 +
 
 +
<cmath>(r+s)^3+(s+t)^3+(t+r)^3=(r+s+t)^3-6rst+(r^3+s^3+t^3)</cmath>
 +
 
 +
Recall that <math>a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)</math>. Thus, we get
 +
 
 +
<cmath>(r+s)^3+(s+t)^3+(t+r)^3=(r+s+t)^3-3rst+(r+s+t)(r^2+s^2+t^2-rs-st-tr)</cmath>
 +
 
 +
Plugging in <math>(r+s+t)=0</math> we get
 +
 
 +
<cmath>(r+s)^3+(s+t)^3+(t+r)^3=0-3rst+0=-3\cdot -251=\boxed{753}</cmath>
 +
 
 +
~firebolt360
  
 +
==Solution 10==
  
 +
<math>8x^3+1001x+2008=0</math>
  
−
Also, Newton's Sums yields an answer through the application.
+
We want to find <math>(r+s)^3+(s+t)^3+(t+r)^3.</math> Let's call this result n.
−
http://www.artofproblemsolving.com/Wiki/index.php/Newton's_Sums
+
 
 +
From vieta's formulas, we find that <math>r+s+t=-0/8=0</math>, <math>rs+st+tr=1001/8</math>, and <math>rst=-2008/8=-251.</math>
 +
 
 +
Expanding and rearranging gives us <math>n=(r+s)^3+(s+t)^3+(t+r)^3=r^3+3r^2s+3rs^2+s^3+s^3+3s^2t+3st^2+t^3+t^3+3t^2r+3tr^2+r^3=2r^3+2s^3+3r^2s+3rs^2+3s^2t+3st^2+3t^2r+3tr^2=3(r^3+s^3+t^3-(r^2s+rs^2+s^2t+st^2+t^2r+tr^2))-(r^3+s^3+t^3)=3((r+s+t)(r^2+s^2+t^2))-((r+s+t)^3-3(r^2s+rs^2+s^2t+st^2+t^2r+tr^2)-6rst)=3(r+s+t)((r+s+t)^2-2(rs+st+tr))-((r+s+t)^3-3(r^2s+rs^2+s^2t+st^2+t^2r+tr^2)-6rst)</math>
 +
 
 +
Let <math>k=r^2s+rs^2+s^2t+st^2+t^2r+tr^2</math>
 +
 
 +
<math>k=r^2s+rs^2+s^2t+st^2+t^2r+tr^2=(r+s+t)(r^2+s^2+t^2)-(r^3+s^3+t^3)=(r+s+t)((r+s+t)^2-2(rs+st+tr))-((r+s+t)^3-3k-6rst)=(0)(0^2-2(1001/8))-(0^3-3k-6(-251))=0-(0-3k+1506)=3k-1506</math>
 +
 
 +
Solving gives us <math>k=753</math>
 +
 
 +
<math>n=3(r+s+t)((r+s+t)^2-2(rs+st+tr))-((r+s+t)^3-3(r^2s+rs^2+s^2t+st^2+t^2r+tr^2)-6rst)=3(0)(0^2-2(1001/8))-(0^3-3k-6(-251))=0-(0-3(753)+1506)=753</math>
 +
 
 +
Therefore, the answer is <math>753.</math>
 +
 
 +
Also note that this is the only solution that still would have worked effectively if <math>r+s+t</math> was nonzero.
  
 
== See also ==
 
== See also ==

Latest revision as of 18:18, 2 January 2024

Problem

Let $r$, $s$, and $t$ be the three roots of the equation \[8x^3 + 1001x + 2008 = 0.\] Find $(r + s)^3 + (s + t)^3 + (t + r)^3$.

Video Solution by Punxsutawney Phil

https://www.youtube.com/watch?v=6mYZYh9gJBs

Solution 1

By Vieta's formulas, we have $r + s + t = 0$ so $t = -r - s.$ Substituting this into our problem statement, our desired quantity is \[(r + s)^3 - r^3 - s^3 = 3r^2s + 3rs^2 = 3rs(r + s).\] Also by Vieta's formulas we have \[rst = -rs(r + s) = -\dfrac{2008}{8} = -251\] so negating both sides and multiplying through by 3 gives our answer of $\boxed{753}.$

Solution 2

By Vieta's formulas, we have $r+s+t = 0$, and so the desired answer is $(r+s)^3 + (s+t)^3 + (t+r)^3 = (0-t)^3 + (0-r)^3 + (0-s)^3 = -(r^3 + s^3 + t^3)$. Additionally, using the factorization \[r^3 + s^3 + t^3 - 3rst = (r+s+t)(r^2 + s^2 + t^2 - rs - st - tr) = 0\] we have that $r^3 + s^3 + t^3 = 3rst$. By Vieta's again, $rst = \frac{-2008}8 = -251 \Longrightarrow -(r^3 + s^3 + t^3) = -3rst = \boxed{753}.$

Solution 3

Vieta's formulas gives $r + s + t = 0$. Since $r$ is a root of the polynomial, $8r^3 + 1001r + 2008 = 0\Longleftrightarrow - 8r^3 = 1001r + 2008$, and the same can be done with $s,\ t$. Therefore, we have \begin{align*}8\{(r + s)^3 + (s + t)^3 + (t + r)^3\} &= - 8(r^3 + s^3 + t^3)\\ &= 1001(r + s + t) + 2008\cdot 3 = 3\cdot 2008\end{align*}yielding the answer $\boxed{753}$.

Also, Newton's Sums yields an answer through the application. http://www.artofproblemsolving.com/Wiki/index.php/Newton's_Sums

Solution 4

Expanding, you get: \[r^3 + 3r^2s + 3s^2r +s^3 +\] \[s^3 + 3s^2t + 3t^2s +t^3 +\] \[r^3 + 3r^2t + 3t^2r +t^3\] \[= 2r^3 + 2s^3 + 2t^3 + 3r^2s + 3s^2r + 3s^2t + 3t^2s + 3r^2t + 3t^2r\] This looks similar to $(r+s+t)^3 = r^3 + s^3 + t^3 + 3r^2s + 3s^2r + 3s^2t + 3t^2s + 3r^2t + 3t^2r + 6rst$ Substituting: \[(r+s+t)^3 - 6rst + r^3+s^3+t^3 = (r + s)^3 + (s + t)^3 + (t + r)^3\] Since $r+s+t = 0$, \[(r+s)^3 + (s+t)^3 + (t+r)^3 = (0-t)^3 + (0-r)^3 + (0-s)^3 = -(r^3 + s^3 + t^3)\] Substituting, we get \[(r+s+t)^3 - 6rst + r^3+s^3+t^3 = -(r^3 + s^3 + t^3)\] or, \[0^3 - 6rst + r^3+s^3+t^3 = -(r^3 + s^3 + t^3) \implies 2(r^3 + s^3 + t^3) = 6rst\] We are trying to find $-(r^3 + s^3 + t^3)$. Substituting: \[-(r^3 + s^3 + t^3) = -3srt = \frac{-2008*3}{8} = \boxed{753}.\]

Solution 5

Write $(r+s)^3+(s+t)^3+(t+r)^3=-(r^3+s^3+t^3)$ and let $f(x)=8x^3+1001x+2008$. Then \[f(r)+f(s)+f(t)=8(r^3+s^3+t^3)+1001(r+s+t)+6024=8(r^3+s^3+t^3)+6024=0.\] Solving for $r^3+s^3+t^3$ and negating the result yields the answer $\boxed{753}.$

Solution 6

Here by Vieta's formulas: $r+s+t = 0$ --(1)

$rst = \frac{-2008}{8} = -251$ --(2)

By the factorisation formula: Let $a = r+s$, $b = s+t$, $c = t+r$, $a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = 0$ (By (1))

So \[a^3+b^3+c^3 = 3abc = 3(r+s)(s+t)(t+r) = 3(-t)(-r)(-s) = 3[-(-251)] = \boxed{753}.\]

Solution 7

Let's construct a polynomial with the roots $(r+s), (s+t),$ and $(t+r)$.

sum of the roots:

$=2(r+s+t)=2\cdot0=0$

pairwise product of the roots:

$(r+s)(s+t)+(s+t)(t+r)+(t+r)(r+s)=r^2+s^2+t^2+3(rs+st+tr)$

$=(r+s+t)^2+rs+st+tr=0+\frac{1001}{8}$

product of the roots:

$(r+s)(s+t)(t+r)=r^2t+r^2s+s^2r+s^2t+t^2r+t^2s+3rst$

$=(rs+st+tr)(r+s+t)-3rst+2rst=-rst=-\frac{2008}{8}$

thus, the polynomial we get is

$x^3+\frac{1001}{8}x+-\frac{2008}{8}=0$

as $(r+s), (s+t),$ and $(t+r)$ are roots of this polynomial, we know that (using power reduction)

$(r+s)^3+\frac{1001}{8}(r+s)-\frac{2008}{8}=0$

$(s+t)^3+\frac{1001}{8}(s+t)-\frac{2008}{8}=0$

$(t+r)^3+\frac{1001}{8}(t+r)-\frac{2008}{8}=0$

adding all of the equations up, we see that

$(r+s)^3+(s+t)^3+(t+r)^3=3\cdot\frac{2008}{8}-\frac{1001}{8}(2r+2s+2t)=251(3)+0=\boxed{753}$

Solution 8

We want to find what is $-(r^3+s^3+t^3)$ which reminds us of Newton sum. So we can see that $8S_3+0\cdot S_2+1001\cdot S_1+3\cdot 2008=0$ Notice that $S_1=0$ so it is just $S_3=-\frac{2008\cdot 3}{8}=-753$, the desired answer is $\boxed{753}$

~bluesoul

Solution 9

This solution uses Vietas, as with everyone else's solution. Expanding the expression we get

\[(r+s)^3+(s+t)^3+(t+r)^3 = r^3+3r^2s+3rs^2+\dots +3s^2t+3ts^2+t^3\]

Seeing the cubes, we try to find a $(r+s+t)^3$ and upon doing so, we get

\[(r+s)^3+(s+t)^3+(t+r)^3=(r+s+t)^3-6rst+(r^3+s^3+t^3)\]

Recall that $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$. Thus, we get

\[(r+s)^3+(s+t)^3+(t+r)^3=(r+s+t)^3-3rst+(r+s+t)(r^2+s^2+t^2-rs-st-tr)\]

Plugging in $(r+s+t)=0$ we get

\[(r+s)^3+(s+t)^3+(t+r)^3=0-3rst+0=-3\cdot -251=\boxed{753}\]

~firebolt360

Solution 10

$8x^3+1001x+2008=0$

We want to find $(r+s)^3+(s+t)^3+(t+r)^3.$ Let's call this result n.

From vieta's formulas, we find that $r+s+t=-0/8=0$, $rs+st+tr=1001/8$, and $rst=-2008/8=-251.$

Expanding and rearranging gives us $n=(r+s)^3+(s+t)^3+(t+r)^3=r^3+3r^2s+3rs^2+s^3+s^3+3s^2t+3st^2+t^3+t^3+3t^2r+3tr^2+r^3=2r^3+2s^3+3r^2s+3rs^2+3s^2t+3st^2+3t^2r+3tr^2=3(r^3+s^3+t^3-(r^2s+rs^2+s^2t+st^2+t^2r+tr^2))-(r^3+s^3+t^3)=3((r+s+t)(r^2+s^2+t^2))-((r+s+t)^3-3(r^2s+rs^2+s^2t+st^2+t^2r+tr^2)-6rst)=3(r+s+t)((r+s+t)^2-2(rs+st+tr))-((r+s+t)^3-3(r^2s+rs^2+s^2t+st^2+t^2r+tr^2)-6rst)$

Let $k=r^2s+rs^2+s^2t+st^2+t^2r+tr^2$

$k=r^2s+rs^2+s^2t+st^2+t^2r+tr^2=(r+s+t)(r^2+s^2+t^2)-(r^3+s^3+t^3)=(r+s+t)((r+s+t)^2-2(rs+st+tr))-((r+s+t)^3-3k-6rst)=(0)(0^2-2(1001/8))-(0^3-3k-6(-251))=0-(0-3k+1506)=3k-1506$

Solving gives us $k=753$

$n=3(r+s+t)((r+s+t)^2-2(rs+st+tr))-((r+s+t)^3-3(r^2s+rs^2+s^2t+st^2+t^2r+tr^2)-6rst)=3(0)(0^2-2(1001/8))-(0^3-3k-6(-251))=0-(0-3(753)+1506)=753$

Therefore, the answer is $753.$

Also note that this is the only solution that still would have worked effectively if $r+s+t$ was nonzero.

See also

2008 AIME II (Problems • Answer Key • Resources)
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